integral (x^2-1)^{1/2}

Lösung

integral

(x^{2} - 1)^{\frac{1}{2}}

\frac{1}{2} x (x^{2} - 1)^{\frac{1}{2}} - \frac{1}{2} ln x + (x^{2} - 1)^{\frac{1}{2}} + C

Schritte zur Lösung

\int (x^{2} - 1)^{\frac{1}{2}} d x

Trigonometrische Substitution anwenden

= \int tan^{2} (u) sec (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int (- 1 + sec^{2} (u)) sec (u) d u

Multipliziere aus

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= \int - sec (u) + sec^{3} (u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int sec (u) d u + \int sec^{3} (u) d u

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= - ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Setze in

u = arcsec (x)

ein

= - ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣ + \frac{1}{2} sec (arcsec (x)) tan (arcsec (x)) + \frac{1}{2} ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣

Vereinfache

- ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣ + \frac{1}{2} sec (arcsec (x)) tan (arcsec (x)) + \frac{1}{2} ln ∣ tan (arcsec (x)) + sec (arcsec (x)) ∣ : \frac{1}{2} x (x^{2} - 1)^{\frac{1}{2}} - \frac{1}{2} ln x + (x^{2} - 1)^{\frac{1}{2}}

= \frac{1}{2} x (x^{2} - 1)^{\frac{1}{2}} - \frac{1}{2} ln x + (x^{2} - 1)^{\frac{1}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x (x^{2} - 1)^{\frac{1}{2}} - \frac{1}{2} ln x + (x^{2} - 1)^{\frac{1}{2}} + C

in t e g r a l \frac{x + 1}{x ^{2} + x + 6}

in t e g r a l ln (1 - z)

in t e g r a l \frac{1}{3 ^{x}}

in t e g r a l e^{- x - 3}

in t e g r a l r^{3}